1896-97.] Prof. Anglin on Circular Functions of 3 0 and 5 6. 455 
Thus 
A ft MP> 
sin 30/sin 0 = y-y -1 = 3- 4 . = 3-4 sin 2 0 
' OC AB 
Aft A M 
cos 30/cos 0 = y-y - 3 = 4- — -3 = 4 cos 2 6 - 3 . 
7 OC AB 
tan 30/tan 0 
AC/OC - 1 
AC/OC - 3 
3MA - MB 3 - tan 2 0 
MA - 3MB = 1 “3 taiD0 ' 
4. Making 
< CPD = < APC, the < PI)B = 5 0 . 
Since the triangles OPD, CPD are similar, we have 
OD A OPD OP 2 
CD - A CPD CP 2 ' 
We shall express each of the ratios sin 50/sin 0 , cos 50/cos 0, etc., 
as a function of AC/OC. 
JSTow 
and 
Thus 
. MP MP AP MP AC 
sm 50 - Dp - Ap . Dp - Ap . . 
AC = AC OC_AC/OD_ \ 
CD' - OC ' CD ~ OCVCD V’ 
OD OP 2 AO 2 . AC CP 
— — — _ Sill CP _ 
CD CP 2 AC . OC ’ “ CP OC 
_ (AC - OC) 2 _ AC OC 0 
AC . OC ~OC + AC 
sin 50/sin 0 = 
AC\ 2 
OC/ 
AC 
OC 
+ 1 
= /apm 
= \AO/ 
Uo; 
+i 
= 16 cos 4 0 — 12 cos 2 0 + 1 = 5 - 20 sin 2 0 + 1 6 sin 4 0 . 
Now 
cos 50 = 
DM^AM AD 
DP “ DP DP ’ 
AM AM AP _ n AC, 
DP - AP ' DP “ cos “ ■ CD ’ 
